A Brief History of Particle Physics (In Tables)

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Hint: use the remainder of the finite Taylor expansion x n~(~) = 1 / ( x - t)gf(g+l)(t)dt. 0 4. Suppose f(x) satisfies the condition of exercise 3 above, then oo f e-"Xf(x)dx ~ ~-'~f(n)(O)w - ~ - 1 as w -+ +oo. ,~ f(x)d x _ ~ 0 f(")(O)n! e-~*x"dx n=0 -- e-~RN(x)dx 0 0 N/(-) (0) n! n=O e-~x~dx. a 5. Watson 's lemma Suppose oo f(t)-- Eant(~-l) , 0_~t_~a+5, a>0, 5>0, n=l and f ( t ) < K e bt f o r t _ > a , K>0, b>0, then o0 oo F(s) - / e - ~ t f ( t ) d t ~ 0 ~-~a"r(n-)s-e'r n=l uniformly valid for Is[ large and [arg s[ _< 2 - A, ~ > O.

Elementary Operations on Asymptotic Expansions where dn(x) is defined by the relations n ao(x)do(x) = 1 and E a m ( x ) d n _ m ( X ) = O, n- 1,2, . . m"-O Consecutively we deal with the nonalgebraic operations of integration and differentation. Because integration is very easy we give first two theorems concerning this operation; the proofs are left to the reader. 4*) Xo where do" is the infinitesimal path element along C. 1) with respect to formulated with the aid of Theorems 4 and 5 respectively.

16) that R(Nm) - o ( ~ g + l ) , uniformly in Ix--xol <_ a implies R(Nm+l) -o(~N+l), uniformly in IX--Xol <_ a; induction on m yields that R(Nm) - - O ( ~ g + l ) uniformly in I x - x01 _< a for each value of m. 17) with M -- N + 1. 32 Chapter 3. Regular Perturbations In~)(x; e) - R(~)(x; e)l -< BeN+l, uniformly for Ix -- x0[ _< a, with B constant independent of e, which we take larger than 1. 17) again we get similarly IR(~)(~; ~)- R(~)(~; ~)l-< nN+~ sup IR(~)(~;6)- R(~)(~; ~)II~- ~ol Ix-xol

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